Macroeconomics
4th Edition
ISBN: 9780393602487
Author: Jones, Charles I.
Publisher: W. W. Norton & Company
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Question
Chapter 6.A, Problem 6E
a)
To determine
Growth rate of output per person along a balanced growth path.
b)
To determine
Level of output per person along a balanced growth path.
b)
To determine
Explain the growth and level of output per person along a balanced growth path with a given geometric series of formula.
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Assume an endogenous growth model with labour augmenting technology. The production function is Y=RK,AN), with A = 2(KIN) such that y = 2k. If the savings rate is s= 0.08, the rate of population
growth is n = 0.03, and the rate of depreciation is d= 0.04, what is the growth rate of output per capita?
Multiple Choice
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Consider the Solow model with a production function Y(t) = A*K(t)αL(t)1-α, Where A is a fixed technological parameter.
Explicitly solve for the steady-state value of the per capita capital stock and per capita income.
How do these values change in response to a rise in (a) the technological parameter A, (b) the rate of saving s, (c) α , (d) δ, the depreciation rate, and the population growth rate n?
Consider the endogenous growth model AK, in which the production function is given by Y = AK. Suppose s denotes the saving rate; that δ represents the depreciation rate; and that the variable that represents the population and that grows at the rate n. Calculate the growth rate of capital per capita in the same way as for the Solow model and, from there, solve the differential equation to obtain the capital per capita (denoted by k) as a function of time.
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- (a) Consider an economy that is initially in a steady state equilibrium. Assume that in this equilibrium it has a saving rate of 50 per cent and a depreciation rate of 2 per cent. Further assume that the population growth rate is 3% and that the level of output produced can be represented by the following production function: = where A = 1 and = 0.5. Use the Solow-Swan model to determine the level of capital per worker and output per worker in this economy. (1 mark) (b) Now suppose the government introduces a set of policies to improve the institutional set up as well as better production technique which increases total factor productivity by double. What is the new steady state level of capital per worker and output per worker? (1 mark) (c) Use a Solow-Swan diagram to show the qualitative effects of this new government policy upon steady state output per worker and capital per worker. Briefly describe the intuition behind this result. (1 mark) (d) Now suppose, population growth rate…arrow_forwardConsider the Solow-Swan growth model, with a savings rate, s, a depreciation rate,8, and a population growth rate, n. The production function is given by: Y = AK + BK¹/2 H¹/4L¹/4 where A and B are positive constants. Note that this production is a mixture of Romer's AK model and the neoclassical Cobb-Douglas production function. (a) Express output per person, y =Y/L, as a function of capital per person, k =K/L.arrow_forwardConsider the continuous-time Solow growth model as discussed in the lecture. The economy is on its balanced growth path with labor augmenting technological progress at rate g and population growth at rate n. The depreciation rate is 8. (a) Derive the dynamic equation of aggregate capital K. (b) Normalize the aggregate capital K by technology A and population L and denote the capital per unit of effective labor as k = K/(AL). Derive the dynamic equation of k .(c) Use the phase diagram to illustrate the steady state level of k and the dynamics of k if k starts from a level lower than the steady state level. (d) Suppose that the economy is on its balanced growth path at time to. Unex- pectedly, the population growth rate drops down to n' <n at to, stays at n' until t1, and resumes to n after t1. How does k respond, between to and t1, and afterwards?arrow_forward
- Problem 1. Consider the Solow-Swan growth model, with a savings rate, s, a depreciation rate, 8, and a population growth rate, n. The production function is given by Y = AK + BK³3/4L1/4 where A and B are positive constants. Note that this production is a mixture of Romer's AK model and the neoclassical Cobb- Douglas production function. • (i) Does this production function exhibit constant returns to scale? Explain why. (ii) Does it exhibit diminishing returns to physical capital? Explain why. • (ii) Express output per person, y =- as a function of capital per person, k =. • (iv) Write down an expression for y/k as a function of k and graph. (Hint: as k goes to infinity, does the ratio y/k approach zero?) (v) Use the production function in per capita terms to write the fundamental equation of the Solow-Swan model. • (vi) Suppose first that sA 8 + n. Draw the savings and depreciation curves, making sure to label the steady state level of capital(if it exists). Under these circumstances,…arrow_forwardProblem 1. Consider the Solow-Swan growth model, with a savings rate, s, a depreciation rate, 8, and a population growth rate, n. The production function is given by Y = AK + BK³3/4L1/4 where A and B are positive constants. Note that this production is a mixture of Romer's AK model and the neoclassical Cobb- Douglas production function. • (i) Does this production function exhibit constant returns to scale? Explain why. (ii) Does it exhibit diminishing returns to physical capital? Explain why. • (ii) Express output per person, y =- -, as a function of capital per person, k =. • (iv) Write down an expression for y/k as a function of k and graph. (Hint: as k goes to infinity, does the ratio y/k approach zero?) (v) Use the production function in per capita terms to write the fundamental equation of the Solow-Swan model. • (vi) Suppose first that sA 8 + n. Draw the savings and depreciation curves, making sure to label the steady state level of capital(if it exists). Under these…arrow_forwardProblem 1. Consider the Solow-Swan growth model, with a savings rate, s, a depreciation rate, 8, and a population growth rate, n. The production function is given by Y = AK + BK³3/4L1/4 where A and B are positive constants. Note that this production is a mixture of Romer's AK model and the neoclassical Cobb- Douglas production function. • (i) Does this production function exhibit constant returns to scale? Explain why. • (ii) Does it exhibit diminishing returns to physical capital? Explain why. (iii) Express output per person, y =, as a function of capital per person, k =. • (iv) Write down an expression for y/k as a function of k and graph. (Hint: as k goes to infinity, does the ratio y/k approach zero?) (v) Use the production function in per capita terms to write the fundamental equation of the Solow-Swan model. • (vi) Suppose first that sA 8 + n. Draw the savings and depreciation curves, making sure to label the steady state level of capital(if it exists). Under these…arrow_forward
- Consider the Solow model without technological progress and an economy with the following production function, Y=A[Kα+Gα]1/α where α<1, K is private capital and G is public capital that is used freely and provided by the government. The level of technology A is fixed and assumed to be equal to 1. A. Does this production function feature constant returns to scale? Explain. In order to finance public capital, the government taxes all investment on private capital at the rate 0<τ<1. So, the revenue raised by the government in each period is sKYt(1−τ) where sK is the private savings rate so that sKYt is pre-tax private savings. Public investment towards public capital is a constant fraction sG of total revenue. Then, the accumulation equations for private and public capital, respectively, are, Kt+1−Kt=sKYt(1−τ)−δKt Gt+1−Gt=sG(sKYtτ)−δGt where δ is the common depreciation rate B. Consider a balanced growth path where the growth rates of private capital is equal to the growth…arrow_forwardIn our one country model of technology growth, y = A(1- γA). Suppose that the country temporarily raises its level of γA. (a) Draw two graphs, one for y and one for A, showing how the time paths of output per worker (y) and productivity (A) will compare under this scenario with what would have happened if there had been no change in γA Please do fast ASAP fastarrow_forward3) There are two countries, Anihc (country A) and Bapan (country B), with the same production function f (k) = 5k0.5. However, country A has saving rates of 0.2, depreciation rate of 0.2 and population growth of 0.2; while country B has saving rates of 0.1, depreciation rate of 0.15 and population growth of 0.05. Using the Solow model: a. Find the steady state capital-labor ratio for each country. b. Find the steady state output per worker, and the steady state consumption per worker for each country. c. Which country produces the most per capita? Which country consumes the most per capita?arrow_forward
- Two countries, X & Y both have the production function f(k) = k0.5 but X starts with an initial k of 4 while Y starts with initial k of 16. Both of the countries have the same values for these exogeneous variables: s = 0.2, A = 1, depreciation rate = 0.05, n = 0, L0 = 10. As you can see, X starts out with just 25% as much capital as Y. How many periods does it take before X has at least 50% as much capital as Y? (A spreadsheet will help a lot with this)arrow_forwardIn our one country model of technology growth, y = A(1- yA). Suppose that the country temporarily raises its level of yA. (a) Draw two graphs, one for y and one for A, showing how the time paths of output per worker (y) and productivity (A) will compare under this scenario with what would have happened if there had been no change in yA.arrow_forwardConsider an economy that has access to a production technology Y = AKαL1−α where Y is output, A is the level of technology, K is capital and L is the amount of labor in the economy. Capital evolves according to K˙ = sY (thus, the depreciation rate δ = 0). The x˙ population growth rate is n. (Throughout, gx = x , where x can be any of the variables in the model.) (a) Assume that technology is determined by A = BKφ What sort of endogenous growth model is this? Find K/K in terms of the K, L, and other parameters of the model.arrow_forward
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